Experimental set up

A rectangular channel set up was used for this case study, this case never flows in floodplain. All the information about the case of study could be consulted here: the paper Proust et al., (2022) and the numerical data Dataset Proust et al., (2024):

Here the features of the experiment:

Main features for the estimation

Laboratory measurements

Check calibration data:

First, a sample of all previously shown measurements was taken to be used in all the tested cases. Then, a guiding question was formulated to trigger the experiment and help provide an answer.

Calibration data

Questions:

To answer all the questions, the experiment below is performed varying the degrees of the Legendre polynomial from 0 to 3.

  1. Could the methodology be used to estimate the friction coefficient of the glass without any prior information?

    Answer: the methodology can estimate the friction using only WSE measurements at a few positions. No prior information is required because friction does not vary in the longitudinal direction, so the WSE profile remains nearly parallel to the riverbed.

  2. What happens if the degree of the Legendre polynomial increases?Could the methodology then estimate more than a single friction constant in longitudinal?

    Answer: increasing the degree of the Legendre polynomial introduces more degrees of freedom, allowing greater spatial variation of friction during calibration. The results indicate that even as the degree increases, the MAP estimation tends to converge toward a constant friction. However, the higher the degree, the more instabilities are observed at the boundaries.

  3. How to select the best estimation considering the Bayesian method for estimating the parameters of a model?

    Answer: the DIC criterion is designed for comparing Bayesian estimations, and previous studies support its use. In this case, the results show that the best estimation corresponds to degree 0, i.e. constant friction along the longitudinal direction, which was expected since the material of the hydraulic flume is glass throughout.

Experiment: 4_WSE_main_channel_real_uncertainty

First, a plot of the DIC criterion will be shown. Then the friction estimations obtained with the MAP estimator, along with the simulated WSE, will be presented for all degrees higher than 0.

DIC criterion

In this case, the best estimation indicates that the friction is constant with any variation from upstream to downstream. That’s why, all calibration results will be presented for this polynomial degree.

Degree 0 of the Legendre polynomial: constant friction

Check MCMC

All MCMC samples

MCMC cooked

Corelation plot of MCMC cooked

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm
a0_min Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2.00100e+03 2001.000000 2001.0000000 2.00100e+03 2001.000000
Minimum 9.54560e+01 2.79900e-04 5.498560 5.5250300 7.99571e+00 8.201460
Maximum 1.00168e+02 1.21710e-03 19.181100 16.1726000 2.52415e+01 26.604700
Range 4.71200e+00 9.37200e-04 13.682500 10.6476000 1.72458e+01 18.403200
Mean 9.77453e+01 6.29700e-04 10.177800 10.2007000 1.52759e+01 15.248600
Median 9.76870e+01 6.19400e-04 9.994480 10.1352000 1.49445e+01 14.868100
Q10% 9.69427e+01 4.68300e-04 7.791380 8.0041100 1.16357e+01 11.520900
Q25% 9.72114e+01 5.28900e-04 8.754500 8.8365600 1.33058e+01 12.944200
Q75% 9.82521e+01 7.21600e-04 11.360200 11.3442000 1.71028e+01 17.250200
Q90% 9.86340e+01 8.08400e-04 12.711000 12.7979000 1.93591e+01 19.604100
St.Dev. 7.16794e-01 1.41400e-04 1.978360 1.8320200 2.96707e+00 3.205620
Variance 5.13793e-01 0.00000e+00 3.913900 3.3562800 8.80352e+00 10.276000
CV 7.33330e-03 2.24511e-01 0.194380 0.1795970 1.94233e-01 0.210224
Skewness 4.51511e-01 4.02530e-01 0.600049 0.4697740 4.78268e-01 0.654463
Kurtosis 6.88676e-01 1.95220e-01 0.629770 -0.0279754 6.15238e-02 0.314980
MaxPost 9.75897e+01 6.04200e-04 9.236950 8.9984800 1.49100e+01 15.529900
a0_min Y1_intercept
St.Dev. 0.716794 0.141369
MaxPost 97.589700 0.604244

Estimation of the friction coefficient

Residuals

In termes of WSE

In termes of discharge

Partial conclusions

  • Using only WSE observations in a rectangular channel, it is possible to accurately and precisely estimate the friction of the hydraulic flume’s material, assuming the friction remains constant along the longitudinal direction.
  • Convergence reached quickly during the exploration of the parameters
  • Any corelation between the parameters of the model and the error model is detected

Degree 1 of the Legendre polynomial

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm
a0_min a1_min Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2001.0000000 2.00100e+03 2001.000000 2001.000000 2001.000000 2001.000000
Minimum 9.57859e+01 -6.7689700 2.52100e-04 4.846070 5.246050 7.584500 7.415690
Maximum 1.00246e+02 9.0211700 1.19870e-03 19.881100 19.985300 31.584100 28.704200
Range 4.46010e+00 15.7901000 9.46600e-04 15.035000 14.739200 23.999600 21.288500
Mean 9.78389e+01 0.8132030 6.46000e-04 10.234300 10.397800 15.231700 15.343000
Median 9.78244e+01 0.7446890 6.26200e-04 10.047400 10.262100 15.132100 14.950600
Q10% 9.69500e+01 -2.0849800 4.80100e-04 7.615360 7.917600 11.295700 11.402300
Q25% 9.73363e+01 -0.7682700 5.53400e-04 8.710490 8.932340 12.775200 12.912000
Q75% 9.82455e+01 2.4414800 7.22900e-04 11.613500 11.694900 17.279500 17.442100
Q90% 9.88864e+01 3.7013700 8.31000e-04 12.994600 13.005800 19.343300 19.639800
St.Dev. 7.34273e-01 2.3339700 1.37700e-04 2.096470 2.060880 3.222170 3.302210
Variance 5.39157e-01 5.4474300 0.00000e+00 4.395200 4.247230 10.382400 10.904600
CV 7.50490e-03 2.8701000 2.13231e-01 0.204848 0.198204 0.211544 0.215226
Skewness 1.47594e-01 -0.0940681 5.32159e-01 0.441203 0.614919 0.437301 0.692990
Kurtosis 1.05091e-02 -0.1046220 5.10203e-01 0.192198 0.853477 0.593110 0.764940
MaxPost 9.76659e+01 1.3908700 5.46900e-04 9.595060 9.887800 12.857100 14.110700
a0_min a1_min Y1_intercept
St.Dev. 0.734273 2.33397 0.137740
MaxPost 97.665900 1.39087 0.546857

Estimation of the friction coefficient

Residuals

In termes of WSE

Degree 2 of the Legendre polynomial

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm
a0_min a1_min a2_min Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2001.0000000 2001.0000000 2.00100e+03 2001.000000 2001.000000 2001.000000 2001.000000
Minimum 9.50853e+01 -7.9169300 -14.9179000 3.37500e-04 5.014510 5.386390 7.890460 7.832800
Maximum 1.00985e+02 8.4200700 17.0317000 1.19230e-03 17.935300 18.371300 24.909100 27.048800
Range 5.89970e+00 16.3370000 31.9496000 8.54700e-04 12.920800 12.984900 17.018600 19.216000
Mean 9.77880e+01 0.9619520 -0.0314878 6.52200e-04 10.205200 10.261400 15.143100 14.997100
Median 9.77258e+01 1.0716300 -0.3644200 6.51100e-04 9.951910 10.102700 14.854500 14.872700
Q10% 9.67616e+01 -2.1062300 -7.4504100 4.84800e-04 7.817560 7.767680 11.424500 11.469300
Q25% 9.71982e+01 -0.7618010 -4.0387600 5.51100e-04 8.602580 8.842100 13.124900 12.802800
Q75% 9.83237e+01 2.5970800 3.9951400 7.38600e-04 11.454400 11.404100 16.894200 16.714500
Q90% 9.89365e+01 4.2098100 7.5742200 8.35300e-04 12.897500 12.974300 19.012700 18.561900
St.Dev. 8.25961e-01 2.4531700 5.7153800 1.41500e-04 2.071270 2.103930 2.993670 2.851680
Variance 6.82211e-01 6.0180500 32.6656000 0.00000e+00 4.290140 4.426520 8.962030 8.132060
CV 8.44640e-03 2.5502000 181.5110000 2.16982e-01 0.202961 0.205033 0.197692 0.190149
Skewness 2.81157e-01 -0.0207898 0.1763540 4.60201e-01 0.643372 0.683571 0.462104 0.419420
Kurtosis 1.79208e-01 -0.1019750 -0.2726330 3.71729e-01 0.453064 0.844478 0.223736 0.460300
MaxPost 9.76545e+01 0.6991390 0.1643870 6.38800e-04 9.070160 9.682910 14.301500 14.518400
a0_min a1_min a2_min Y1_intercept
St.Dev. 0.825961 2.453170 5.715380 0.141518
MaxPost 97.654500 0.699139 0.164387 0.638756

Estimation of the friction coefficient

Residuals

In termes of WSE

Degree 3 of the Legendre polynomial

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm
a0_min a1_min a2_min Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2001.0000000 2001.0000000 2.00100e+03 2001.000000 2001.000000 2001.000000 2001.000000
Minimum 9.50853e+01 -7.9169300 -14.9179000 3.37500e-04 5.014510 5.386390 7.890460 7.832800
Maximum 1.00985e+02 8.4200700 17.0317000 1.19230e-03 17.935300 18.371300 24.909100 27.048800
Range 5.89970e+00 16.3370000 31.9496000 8.54700e-04 12.920800 12.984900 17.018600 19.216000
Mean 9.77880e+01 0.9619520 -0.0314878 6.52200e-04 10.205200 10.261400 15.143100 14.997100
Median 9.77258e+01 1.0716300 -0.3644200 6.51100e-04 9.951910 10.102700 14.854500 14.872700
Q10% 9.67616e+01 -2.1062300 -7.4504100 4.84800e-04 7.817560 7.767680 11.424500 11.469300
Q25% 9.71982e+01 -0.7618010 -4.0387600 5.51100e-04 8.602580 8.842100 13.124900 12.802800
Q75% 9.83237e+01 2.5970800 3.9951400 7.38600e-04 11.454400 11.404100 16.894200 16.714500
Q90% 9.89365e+01 4.2098100 7.5742200 8.35300e-04 12.897500 12.974300 19.012700 18.561900
St.Dev. 8.25961e-01 2.4531700 5.7153800 1.41500e-04 2.071270 2.103930 2.993670 2.851680
Variance 6.82211e-01 6.0180500 32.6656000 0.00000e+00 4.290140 4.426520 8.962030 8.132060
CV 8.44640e-03 2.5502000 181.5110000 2.16982e-01 0.202961 0.205033 0.197692 0.190149
Skewness 2.81157e-01 -0.0207898 0.1763540 4.60201e-01 0.643372 0.683571 0.462104 0.419420
Kurtosis 1.79208e-01 -0.1019750 -0.2726330 3.71729e-01 0.453064 0.844478 0.223736 0.460300
MaxPost 9.76545e+01 0.6991390 0.1643870 6.38800e-04 9.070160 9.682910 14.301500 14.518400
a0_min a1_min a2_min Y1_intercept Y2_intercept
St.Dev. 0.825961 2.453170 5.715380 0.141518 2.07127
MaxPost 97.654500 0.699139 0.164387 0.638756 9.07016

Estimation of the friction coefficient

Residuals

In termes of WSE